Numerical Analysis 2011160001 Yu Jieun

01 02 03 Index SIR model Finite Difference Method, FDE Implicit Explicit Crank-Nicolson MATLAB Code 03 Monte Carlo Simulation MATLAB Code

01. SIR model Mathematical modeling of spread of disease

01 Variables ① S = S(t) : the number of susceptible individuals Independent variable : time t Dependent variables ① S = S(t) : the number of susceptible individuals ② I = I(t) : the number of infected individuals ③ R = R(t) : the number of removed individuals

01 Assumptions We can ignore birth and disease-unrelated death, like immigration, since the rate of infection and recovery is much faster than the time scale of births and deaths. The population is therefore closed, of constant size N, 𝑁=𝑆 𝑟 +𝐼 𝑟 +𝑅 𝑟

01 Assumptions Each infected individual has a fixed number 𝛽 of contacts per day that are sufficient to spread the disease and a fixed fraction 𝛾 of the infected group will recover during any given day. 𝑑𝑆 𝑑𝑟 =−𝛽𝐼𝑆, 𝑑𝐼 𝑑𝑟 =𝛽𝐼𝑆−𝛾𝐼, 𝑑𝑅 𝑑𝑟 =𝛾𝐼

01 Assumptions These processes which occur simultaneously are referred to as ‘the Law of Mass Action’, that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned.

01 Suspectible Infective S I R Removed 𝛽𝐼𝑆 𝛾𝐼

01 MATLAB Code

01 General graph of SIR model

02. Finite Difference Model

02 Finite Difference Method Black & Scholes modelled stochastic differential equation about the price which follow Geometric Brownian Motion. where V is the price of the option as a function of stock price S and time t, r is the risk-free interest rate, 𝜎 is the volatility of the stock.

02 Finite Difference Method FDM is a method of calculating a numerical approximation to the partial differential equations to differential equations. It refers to the differential equation which is approximated to the derivative of a continuous function in the following forms. Forward Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥+ℎ,𝑡 −𝑢(𝑥,𝑡) ℎ +𝑂 ℎ Backward Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥,𝑡 −𝑢(𝑥−ℎ,𝑡) ℎ +𝑂 ℎ Central Difference : 𝑢 𝑥 𝑥,𝑡 = 𝑢 𝑥+ℎ,𝑡 −𝑢(𝑥−ℎ,𝑡) 2ℎ +𝑂 ℎ 2 .

02 Finite Difference Method For this, we constitutes a two-dimensional grid consisting of a variable of a partial differential equation.

02 1. Explicit FDM If we use the forward difference at time and a second-order central difference for the space derivative at position 𝑥 𝑗 . we get the recurrence equation:                                                                                We can obtain     from the other values this way :                                                                                     where                       

02 2. Implicit FDM If we use the backward difference at time and a second-order central difference for the space derivative at position 𝑥 𝑗 . we get the recurrence equation: We can obtain from solving a system of linear equations:

02 3. Crank-Nicolson method Finally if we use the central difference at time and a second-order central difference for the space derivative at position we get the recurrence equation:                                                                                                                                                      We can obtain    from solving a system of linear equations:                                                                  

02 MATLAB Code – Implicit FDM , call option

02 MATLAB Code

03. Monte Carlo Simulation

03 Monte Carlo Simulation Monte Carlo Simulation is a problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.

03 MATLAB Code The following MATLAB scripts performs the Monte-carlo simulation.

03 MATLAB Code

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